Kropki Sudoku

Kropki Sudoku — kropki is simply Polish for "dots" — looks like an ordinary 9×9 grid sprinkled with tiny white and black circles sitting on the lines between cells. Those dots are the whole puzzle. Each one reveals a precise numerical relationship between the two cells it touches, and a good Kropki gives you almost no starting digits at all: the dots, read carefully, are enough to solve the entire grid by pure logic.

The two dots and what they mean

Everything in Kropki rests on two symbols, and it pays to learn them cold before anything else:

• A white dot between two cells means the digits are consecutive — they differ by exactly 1, in either order. So a white dot joins pairs like 4 and 5, or 8 and 7.
• A black dot means one digit is exactly double the other — a ratio of 1 to 2. That covers 1–2, 2–4, 3–6 and 4–8, again in either direction.

Reading a single dot

A dot is not just a fact to check at the end — it is a candidate filter you apply the moment you see it. Ask what pairs a dot even allows, and you often shrink two cells at once.

A white dot permits only consecutive pairs: 1–2, 2–3, 3–4, 4–5, 5–6, 6–7, 7–8 and 8–9. Notice that every digit except the extremes has two consecutive neighbours, but 1 has only 2 and 9 has only 8. A black dot is even choosier: the only doubling pairs on a 1–9 grid are 1–2, 2–4, 3–6 and 4–8. That means a cell on a black dot can only ever be one of 1, 2, 3, 4, 6 or 8 — the digits 5, 7 and 9 can never sit beside a black dot, because none of them has a half or a double inside 1–9.

The most powerful reading: the negative constraint

In the standard, "full" version of Kropki, every valid relationship is marked. That turns the empty gaps into information too: if there is no dot between two adjacent cells, then those cells are neither consecutive nor in a 1-to-2 ratio. This "negative constraint" is where most of a hard Kropki actually gets solved. A bare edge next to a 4, for instance, forbids 3, 5, 2 and 8 in the neighbour — four digits gone from a single missing dot.

Solving strategy: start at the extremes

The fastest way into a Kropki is to hunt the edges of the number range, where choices collapse to one. Because 9 has no double and only one consecutive partner, a white dot on a 9 forces its neighbour to 8, and a black dot can never touch a 9 at all. The digit 1 behaves the same way at the bottom: a black dot beside a 1 forces a 2, since 1's only doubling partner is 2.

From those forced cells the ordinary Sudoku rules take over. Once the 8 next to the 9 is placed, it rips through its row, column and box like any given — and because Kropki grids are built to cascade, one forced edge frequently unlocks a whole corner.

A worked example, step by step

Watch how two dots chain together. Here three cells sit in a row: the first is a given 4, joined to the second by a white dot, and the second to the third by a black dot.

Step 1 — read the white dot. The white dot means the second cell is consecutive with 4, so it is either 3 or 5.

Step 2 — the black dot decides. The black dot needs the second cell to have a double or half on the grid. A 5 has neither (10 and 2.5 are both off the grid), so the second cell must be 3 — and its double, 6, fills the third cell.

That is the rhythm of Kropki: each dot is a small equation, and chaining them across a box turns a nearly blank grid into a single forced solution.

Common mistakes

• Reading a dot as an order. Dots never say which side is larger — a white dot on 4 and 5 could be 4-then-5 or 5-then-4. Only other constraints decide direction.
• Forgetting 1–2 is both. The pair 1 and 2 is consecutive and a 1-to-2 ratio, so a dot between them could in principle be drawn either colour; setters pick one, but keep both readings in mind when a cell is 1 or 2.
• Ignoring the missing dots. In full Kropki the absence of a dot is as strong as its presence. Solvers who only "use" the printed dots stall on exactly the puzzles the negative constraint is meant to crack.
• Placing a 5, 7 or 9 on a black dot. None of them has a half or double within 1–9 — a black dot rules them out on sight.

Can I play it here?

Not yet — Kropki Sudoku isn't one of the puzzle types SudokuStreak generates for now. But you can learn its rules above, and there's plenty here to sink your teeth into meanwhile:

• Try the closely related X Sudoku, which you can play right now.
• Browse all the Sudoku variants we do offer.
• Or print free puzzles from the printable PDF page.